jmc

Note that $\gcd (168,693)=21$. Since $\gcd (a,b)=168=8\cdot 21$, both $a$ and $b$ are divisible by 21. Since $\gcd (a,c)=693=21\cdot 33$, both $a$ and $c$ are divisible by 21. Therefore, $\gcd (b,c)$ must be at least 21.

If we take $a=5544$ (which is $21\cdot 8\cdot 33$), $b=168$, and $c=693$, then $\gcd (a,b)=\gcd (5544,168)=168$, $\gcd (a,c)=\gcd (5544,693)=693$, and $\gcd (b,c)=\gcd (168,693)=21$, which shows that the value of 21 is attainable. Therefore, the smallest possible value of $\gcd (b,c)$ is $\boxed{21}$.